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Here's some equations:

1. Orbital Period:

P=G*r^((1-e)/2) where G=force of gravity, r=average orbit radius, and e=gravity exponent

2. Velocity:

V=(G^(1/2))*r^((e+1)/2) (seems to fit for e=-2 and e=1)

I hope I got this right. I used trial and error to figure these equations out.

Well, I got nothing to say but DAAAANG son! Nice!

Here's an example using these equations:
In this case, G seems to equal 7/7952400=8.802374126*10^-7, mainly because the orbital velocity is sqrt(G)*M*m*r^((e+1)/2)=2*sqrt(7)=2*2.6457513...=5.2915026... where e=-1.

Some strange things that I've noticed:

e=1: Orbital period is constant, no matter what the orbital radius is.
e=-1: Orbital velocity is constant and only related to the combined mass.
e>-1: Orbital velocity increases with distance.
e>1: Orbital period decreases with increasing distance.
e=-100: Close objects pull on each other with unimaginable force.
e=100: Objects cannot leave a certain area without going at crazy speeds.
e>1: Escape velocity is infinite; all objects are bound to one another no matter how fast they go.
e=1.1: P=0.5 orbit is 1,048,576 times more distant than P=1 orbit.

Wow, nice work! I finally got a moment to sit down and check out what you'd made -- I'm impressed. Now, you say you used trial and error to do this? That's crazy! I'd love to hear more about the process, since these are not exactly trivial equations. How many data points did you snag, that kinda thing.

I took a look over your equations, and they're *basically* correct.

1. The Orbital Period -- you've got the right dependency on the radius. The placement of 'G' isn't quite right, though. (As a side note, I'm interpreting your 'G' to be G*M, where G is the gravitational constant, and M is the mass of the central star, say.) Your current equation seems to say that the more massive the star, the longer the orbit will take. Take another look at that.

(Also, when I wrote out this equation, there's an extra 2*PI multiplied in. I'm not going to begrudge you a constant, though. The units in my simulator are confusing enough as it is -- there may well be an extra constant thrown into your experimental results due to the framerate or some other such thing.)

2. Velocity -- nailed it. This is indeed the velocity you'd need to have a circular orbit.

Stargate38 wrote:e=1: Orbital period is constant, no matter what the orbital radius is.

One year on Jupiter is the same as a year on Earth! I wonder how far out you'd have to go before planets need to orbit at relativistic speeds.

testtubegames wrote:One year on Jupiter is the same as a year on Earth! I wonder how far out you'd have to go before planets need to orbit at relativistic speeds.

Well that's an easy one. And a hard one at the same time.
On the easy part, we got the formula.
On the hard part, in the gravity simulator universe, there is no relativity. Light moves infinitely fast and such. So sadly, no way to tell what a "relativistic speed" is.

Oops. I forgot to put a ^-1 on the M. Here's a fixed equation:

P=(1/G^n)*2pi*r^((1-e)/2), where n is the correct exponent for this equation to work (still trying to find the right n value, probably 0<n<2)

For e=-3, orbital velocity is basically reduced to 1/r. Since doubling the distance divides the force of gravity by 8, this means that the velocity is basically equal to the force times the area of the orbit (for a circle), because doubling the distance results in half the velocity. Assume that v=1 and G=1 at r=1, v=2 and G=8 at r=0.5. In that case, r=0.5 means the area is only 0.25A, where A=pi*r^2, and v=8*0.25=2. I'm still trying to find the equation for escape velocity. It's infinite for e=-1, but it's equal to the orbital velocity for e=-3, and less than the orbital velocity for e<-3, so I'm considering esc=v*sqrt(-2/(1+e)) as a possibility.

Test for e=-3, v=1, esc=1 (slowed the 2nd and 3rd bodies by a small amount to prevent them from escaping, they'll eventually hit the sun):
Test for e=-9, v=1, esc=0.5 (same as above, but 1st body eventually escapes due to software limitations):
The equation seems to be holding so far.

If (for e=1) c were to equal 299792458, then objects wouldn't be able to orbit beyond a certain distance. For e=100, the problem is even worse, as an object would only have to be 1.2 times as far away to start feeling relativistic effects from the immense gravity. For e=-100, every object with mass would become a black hole due to the same problem.